

A000408


Numbers that are the sum of three nonzero squares.


63



3, 6, 9, 11, 12, 14, 17, 18, 19, 21, 22, 24, 26, 27, 29, 30, 33, 34, 35, 36, 38, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 56, 57, 59, 61, 62, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 81, 82, 83, 84, 86, 88, 89, 90, 91, 93, 94, 96, 97, 98, 99, 101, 102, 104
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OFFSET

1,1


COMMENTS

From Jonathan Vos Post, Mar 03 2010: (Start)
Catalan conjectured that three times any odd square not divisible by 5 is a sum of squares of three primes other than 2 and 3 (regarding 1 as a prime). Catalan stated and Realis proved that every power of 3 is a sum of three squares relatively prime to 3. [See Dickson]
From the abstract of the Berkovich and Jagy paper: "Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n)  ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants p^2 and 16p^2. To prove this identity we employ the SiegelWeil and the SmithMinkowski product formulas."
(End)
a(n) not equal 7 mod 8.  Boris Putievskiy, May 05 2013
A025427(a(n)) > 0.  Reinhard Zumkeller, Feb 26 2015
According to HalterKoch (below), a number n is a sum of 3 squares, but not a sum of 3 nonzero squares (i.e., is in A000378 but not A000408), if and only if it is of the form 4^j*s, where j >= 0 and s in {1,2,5,10,13,25,37,58,85,130,?}, where ? denotes at most one unknown number that, if it exists, is > 5*10^10.  Jeffrey Shallit, Jan 15 2017


REFERENCES

L. E. Dickson, History of the Theory of Numbers, vol. II: Diophantine Analysis, Dover, 2005, p. 267.
Savin Réalis, Answer to question 25 ("Toute puissance entière de 3 est une somme de trois carrés premiers avec 3"), Mathesis 1 (1881), pp. 8788. (See also p. 73 where the question is posed.)
Franz HalterKoch, Darstellung natürlicher Zahlen als Summe von Quadraten, Acta Arithmetica 42 (1982), 1120.


LINKS

Ray Chandler, Table of n, a(n) for n = 1..10000
Alexander Berkovich, Will Jagy, On representation of an integer as the sum of three squares and the ternary quadratic forms with the discriminants p^2, 16p^2, arXiv:1101.2951 [math.NT], 2011.
B. Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), J. Int. Seq. 16 (2013) #13.6.4
S. Mezroui, A. Azizi, M'hammed Ziane, On a Conjecture of Farhi , J. Int. Seq. 17 (2014) #14.1.8
Index entries for sequences related to sums of squares


FORMULA

a(n) = 6n/5 + O(x/sqrt(log n)). (Can the error term be improved?)  Charles R Greathouse IV, Mar 14 2014


MAPLE

N:= 1000: # to get all terms <= N
S:= series((JacobiTheta3(0, q)1)^3, q, 1001):
select(t > coeff(S, q, t)>0, [$1..N]); # Robert Israel, Jan 14 2016


MATHEMATICA

f[n_] := Flatten[Position[Take[Rest[CoefficientList[Sum[x^(i^2), {i, n}]^3, x]], n^2], _?Positive]]; f[11] (* Ray Chandler, Dec 06 2006 *)
pr[n_] := Select[ PowersRepresentations[n, 3, 2], FreeQ[#, 0] &]; Select[ Range[104], pr[#] != {} &] (* JeanFrançois Alcover, Apr 04 2013 *)
max = 1000; s = (EllipticTheta[3, 0, q]  1)^3 + O[q]^(max+1); Select[ Range[max], SeriesCoefficient[s, {q, 0, #}] > 0 &] (* JeanFrançois Alcover, Feb 01 2016, after Robert Israel *)


PROG

(PARI) is(n)=for(x=sqrtint((n1)\3)+1, sqrtint(n2), for(y=1, sqrtint(nx^21), if(issquare(nx^2y^2), return(1)))); 0 \\ Charles R Greathouse IV, Apr 04 2013
(PARI) is(n)= my(a, b) ; a=1 ; while(a^2+1<n, b=1 ; while(b<=a && a^2+b^2<n, if(issquare(na^2b^2), return(1) ) ; b++ ; ) ; a++ ; ) ; return(0) ;
for(n=3, 1e3, if(is(n), print1(n, ", "))); \\ Altug Alkan, Jan 18 2016
(Haskell)
a000408 n = a000408_list !! (n1)
a000408_list = filter ((> 0) . a025427) [1..]
 Reinhard Zumkeller, Feb 26 2015


CROSSREFS

Cf. A004214 (complement), A024795 (numbers with multiplicity), A000404, A000378, A025427.
Sequence in context: A329511 A065940 A024795 * A025321 A153238 A230193
Adjacent sequences: A000405 A000406 A000407 * A000409 A000410 A000411


KEYWORD

nonn


AUTHOR

N. J. A. Sloane and J. H. Conway


STATUS

approved



